Group isomorphism of intersections

manpreet Best Answer 2 years ago

In the syllabus for my course on group theory, there is the following theorem which is proven: "If H⊂ZnH⊂Zn is a subgroup, then H≅ZkH≅Zk for 0≤k≤n0≤k≤n".

I understand most of the proof, but there's a simple line that is intuitive, yet doesn't mathematically make sense to me.

In the proof, which is done by induction, the following homomorphism is defined

The kernel of the map is considered, which are all vectors of the form (m1,...,mn,0)(m1,...,mn,0).

Then the syllabus reads "Since H⊂Zn+1H⊂Zn+1 is a subgroup, also H∩ Ker